Global Analysis of a Delayed Impulsive Lotka-Volterra Model with Holling III Type Functional Response

Abstract

A delayed impulsive Lotka-Volterra model with Holing III type functional response was established. With the help of Mawhin’s Continuation Theorem in coincidence degree theory, a sufficient condition is found for the existence of positive periodic solutions of the system under consideration. By applying the comparison theorem and constructing a suitable Lyapunov functional, the permanence and global attractivity of the model are proved. Two numerical simulations are also given to illustrate our main results.

1. Introduction

Recently, many complicated but realistic predator-prey systems based on classical Holling type functional responses have been analyzed by ecologists and mathematicians; see papers [1–6] and so forth. For example, a positive periodic solution to a Lotka-Volterra model with mutual interference and Holling III type functional response was proposed by Lv and Du in [2]; Zhang and his coworkers studied positive periodic solutions in a predator-prey model with Hassell-Varley type functional response, nonselective harvesting, and multiple delays in paper [6]. The theoretical values of these studies not only have great significance in biological economics but also provide strong support for the management and development of renewable energy.

Hassell [7] introduced the following predator-prey system with mutual interference : After that more scholars have further conducted research. For example, Du and Lv investigated a Lotka-Volterra model with mutual interference and time delays in [8]: Some criteria on the permanence and global attractivity of the above system are found. As far as we know, delay models have been studied and applied extensively in biology, physics, population dynamics, and other fields. However, the assumption of these models with constant environment is rarely the case in real life. A system must be nonautonomous if the environmental fluctuation is taken into account, such as seasonal effects of weather, food supplies, and harvesting. Therefore, it is rational to consider the ecosystem with periodic or almost periodic coefficients. On the other hand, there exist a few discontinuous and impulsive phenomena; for instance, many species are given birth seasonally. If we introduce these impulsive factors into the systems, it is more realistic to analyze the ecology models. For example, Wang and Zhu [4] considered a delayed impulsive prey-predator system with mutual interference:

But as far as we all know, there are few results on the existence and global attractivity of positive periodic solutions of model (1) with delays and impulses. Motivated by these facts, we formulate a delayed impulsive Lotka-Volterra model with Holing III type functional response with initial conditions where and represent the regular harvest or death from spraying pesticide of the predator and prey at time , . In this paper, , , and denote integers, positive integers, and real numbers, respectively, and ; , , and are continuously nonnegative periodic functions with period ; are nonnegative and continuously differentiable periodic functions with period on , and ; and are positive constants and ; and satisfy .

Furthermore, suppose that the following conditions hold.(H1)There exists an integer , such that , , and are fixed points with .(H2), are constants, and ,

This paper is organized as follows. In Section 2, we introduce some useful lemmas. In Section 3, we not only prove the existence of periodic solutions but also study the permanence and the global attractivity of system (4). In Section 4, two examples are given to illustrate the feasibility of our results by using simulation. The last section is a brief conclusion.

2. Preliminaries

Under the assumptions (H1) and (H2), we consider a new system as follows: with initial conditions where

Lemma 1. Suppose that (H1) and (H2) hold; then we have the following.(i)If is a solution of systems (6) and (7), then is a solution of systems (4) and (5), where (ii)If is a solution of systems (4) and (5), then is a solution of systems (6) and (7), where

Proof. (i) For any , , we have Similarly, we have On the other hand, for any , by definition, we obtain that and , . Then we get , Hence, is a solution of systems (4) and (5).(ii) and are continuous on each interval . Since and , we have And combining we know that and are continuous on interval .
Similarly, Therefore, is a solution of systems (6) and (7).

From Lemma 1, we notice that if we want to discuss the existence of an -periodic solution of systems (4) and (5), we only need to discuss the existence of an -periodic solution of systems (6) and (7).

Let and be two normed linear spaces, let be a linear map, and let be a continuous map. If and is closed in , then is called Fredholm operator. If is a Fredholm operator with index zero, there exist continuous projections and such that and Hence, has an inverse mapping . The mapping is called -compact on , if is an open bounded subset, and    is bounded, is compact. Since is isomorphic to , there exists an isomorphism .

Lemma 2 (see [9], (Continuation Theorem)). Let both and be Banach spaces; is a Fredholm operator with index zero; continuous projection is called -compact on , where is an open bounded subset on . If all the following conditions hold,(1)for any , each solution of satisfies ,(2)any , ,(3), where is an isomorphism, then the equation has at least one solution in .

Lemma 3 (see [10]). If , , and , where is a positive constant, then

Lemma 4 (see [11]). If with and for any , then function has a unique inverse satisfying with , for .

3. Main Results

In order to express the formulas conveniently, we introduce a few concepts where is a periodic function with period .

Let be an arbitrary positive solution of systems (6) and (7), for all . Set and . Consider the following model:

Apparently, if system (19) has an -periodic solution , then is an -periodic solution of systems (6) and (7). Hence, we only need to show that system (19) has an -periodic solution.

Set where and the norm , Then both and are Banach spaces.

Define operators , , and as follows, respectively: Define , satisfying Note that , , and is closed in and and are continuous maps satisfying and Hence, is a Fredholm operator with index zero. It implies that has a unique inverse . So we have By a straightforward calculation, we get where By the Lebesgue Convergence Theorem, it is not difficult to notice that and are continuous. By applying the Arzela-Ascoli Theorem, we know that the operator is bounded and is compact, for any open set . Therefore, is -compact on .

In order to use Lemma 2, we need to find an appropriate open and bounded set .

Theorem 5. Assume the following.
(i)  , where , .
(ii) The following algebraic equation set has finite solutions , , and then systems (6) and (7) have at least one -periodic solution.

Proof. Considering the operator equation , , we have Integrating (27) on the interval , we have yielding In view of Lemma 4, we obtain that which together with (28) give From (31), we have where .
Multiplying the first equation of system (27) by and the second one by and integrating them on , we get which imply that If , then and , which implies that there must be a constant such that From (33) and (37), then there exists a constant such that .
On the other hand, it follows from (35) and the Hölder inequality that that is Since and , then and .
From (32), , that is, If , then , , and .
From (31), we obtain that that is, Together with (33) and (41), we notice that there exists a constant such that Furthermore, according to the condition and (32), we have then that is We can find a constant such that Hence, (19), (44), and (48) combine together to make us know that Consider the algebraic equation set From the assumption, there exist finite solutions , , of the above system. By simple computing, we can get its Jacobian matrix obviously .
Set and , , where is a large enough number satisfying . If , then satisfies and , where is an isomorphism. Hence, is a bounded open set. System (19) has at least one -periodic solution in ; that is, system (6) has at least one -periodic solution .

Corollary 6. Suppose that Theorem 5 holds; system (4) also has at least one -periodic solution .

Now, we discuss the permanence of models (6) and (7). Before the main results, we give the definition of permanence.

Definition 7. System (6) is permanent, if there exist positive constants , and time such that any solution of system (6) with initial condition (7) satisfies for all .
Denote

Theorem 8. If , then system (6) is permanent.

Proof. From the first equation of model (6), we have , integrating it on interval , then we get According to Lemma 3, we have , , and then there exists a sufficiently large time such that for all .
At the same time, we can find time such that for .
From the second equation of model (6), we have that is, By solving equation, we obtain that then there must be such that for all , where is a constant.
Further, there exists time such that for all . In view of the first equation of model (6), we have Assume that is local minimal value of system (59); then . Thus, we have that that is, Integrating (59) on and noticing that we can easily get that Thus, there must be a large enough time such that for .
In view of the second equation of system (6), we have Similarly, by solving the equation, we obtain that For the above constant , we can seek time such that, for , In summary, and for Therefore, system (6) is permanent.

Theorem 9. If , then system (4) is permanent and enters eventually into the region , where

Corollary 10. If , then systems (6) and (7) have a positive -periodic solution.

Definition 11. System (6) is globally attractive, if for any two positive solutions and of systems (6) and (7).

Theorem 12. If system (6) satisfies , , and , then system (6) is globally attractive, where

Proof. Assume and are two positive solutions of systems (6) and (7). According to Theorem 8, there exist two positive constants , and such that, for , Define a function By calculating its upper right derivative along the solution of system (6), we get that where Thus, for , where Define further For , yielding that Further define Then we choose the Lyapunov functional as follows: then Since , then hence, we have It follows from and that for Integrating both sides of the above inequality on , we have Since and are uniformly continuous on , by Barbalat’s Lemma [12], we obtain that that is, Therefore, system (6) is globally attractive. Together with that is, , system (4) is globally attractive.

Corollary 13. Let , where is a nonnegative constant. If system (6) satisfies , , and , then system (6) has a unique positive almost -periodic solution which is globally attractive.

4. Numerical Simulation

Example 14. Assume . Considering the following model as application: corresponding to model (87), we take By direct computation, we have , which satisfies the conditions of Theorem 5 and Corollary 6. We can see that system (87) has a positive periodic solution (see Figures 1 and 2).

(a)

(b)

Figure 1 

The integral curves of prey-time and predator-time.

(a)

(b)

Figure 2 

The orbits of prey-predator and time-prey-predator.

Example 15. Assume . Considering the following model as application corresponding to model (89), we take By direct computation, we have , , and . We can see that system (89) is permanent and has a unique positive -periodic solution, which is globally attractive (see Figures 3 and 4). It is easy to verify the accuracy of Theorems 8, 9, and 12 and Corollaries 10 and 13.

(a)

(b)

Figure 3 

The integral curves of prey-time and predator-time.

(a)

(b)

Figure 4 

The orbits of prey-predator and time-prey-predator.

5. Conclusion

In the study of population dynamics, we focus on two aspects: (1) the time-varying evolution of the population and (2) how to implement manual intervention to protect, develop, and utilize the population. Precisely, these two issues are reflected in our model. Regarding the first aspect, we take into account the impact of limited resources on population size; that is, its density has a restriction on the growth of the population size. Hence, we not only use the interspecific growth terms and to reflect the model, where and are delays, but also consider the interference constant and Holling III type functional response. Taking into account the second aspect, we propose a regular pulse harvest in the model. The corresponding ecological system of the model we considered is more complex and has practical significance.

In this paper, we analyze the existence and global attractivity of positive periodic solutions of a delayed impulsive Lotka-Volterra model with Holing III type functional response. We propose two delays and impulses to describe the model. From Theorem 5, we can conclude that the positive periodic solutions of system (4) are delay dependent. This is different from these results that the positive periodic solutions are delay independent, and our conclusion is more general. Furthermore, we have shown the permanence and global attractivity of system (4) under certain conditions. We have found that system (4) has a unique and globally attractive periodic solution, but how can we prove it? We leave it as our work in the future. In addition to delayed and impulsive biological systems, we hope that our analysis can provide valuable design insights and supports to future biological works.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61174209) and the Basic Theory Research Foundation for Engineering Research Institute of USTB (YJ2012-001).